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3L 4s: Difference between revisions

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| Pattern = LsLsLss
| Pattern = LsLsLss
}}
}}
{{MOS intro}}
== Name ==
[[TAMNAMS]] suggests the temperament-agnostic name '''mosh''' for this scale, adopted from an older [[Graham Breed's MOS naming scheme|mos naming scheme]] by [[Graham Breed]]. The name is a contraction of "mohajira-ish".
== Scale properties ==
{{TAMNAMS use}}


{{MOS intro}}
=== Intervals ===
==Name==
{{MOS intervals}}
[[TAMNAMS]] suggests the temperament-agnostic name '''mosh''' for this scale, adopted from an older [[Graham Breed's MOS naming scheme|MOS naming scheme]] by Graham Breed. The name is a contraction of "mohajira-ish".
 
==Notation==
=== Generator chain ===
:''This article assumes [[TAMNAMS]] for naming step ratios, intervals, and scale degrees, and [[Diamond-mos notation|diamond-MOS notation]] for note names.''
{{MOS genchain}}
===Intervals and degrees===
Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.


Being a moment-of-symmetry scale, every [[interval class]] of 3L 4s, except for the unison and octave, has two [[Interval variety|varieties]] – large and small – whose [[Interval quality|relative qualities]] are denoted as major or minor, or augmented, perfect, and diminished for the generators.
=== Modes ===
{| class="wikitable"
{{MOS mode degrees}}
|+
Interval varieties of 3L 4s
! rowspan="2" |Interval class
! colspan="2" |Large variety
! colspan="2" | Small variety
|-
!Size
!Quality
!Size
! Quality
|-
|'''0-mosstep (unison)'''
|0
|Perfect
| 0
|Perfect
|-
| 1-mosstep
|L
|Major
|s
|Minor
|-
| 2-mosstep
|L + s
| Perfect
|2s
|Diminished
|-
|3-mosstep
|2L + s
|Major
|L + 2s
| Minor
|-
|4-mosstep
| 2L + 2s
|Major
| L + 3s
| Minor
|-
|5-mosstep
|3L + 2s
|Augmented
|2L + 3s
| Perfect
|-
| 6-mosstep
|3L + 3s
| Major
|2L + 4s
|Minor
|-
|'''7-mosstep (octave)'''
|3L + 4s
|Perfect
|3L + 4s
|Perfect
|}
===Note names===
For this article, note names are based on diamond-MOS notation, where the naturals JKLMNOP are applied to the step pattern sLsLsLs and the accidentals & (pronounced "am" or "amp") and @ (pronounced "at") are used to represent sharps and flats respectively. Thus, the basic gamut for 3L 4s is the following:


{{MOS gamut}}
=== Proposed names ===
The first set of mode nicknames was coined by [[Andrew Heathwaite]]. The other set was coined by [[User:CellularAutomaton|CellularAutomaton]] and follows the diatonic modes' naming convention by using ancient Greek toponyms that sound similar to Heathwaite's names. The third shows which modes are a mixture of which diatonic modes, as discussed in [[#Theory]].
{{MOS modes
| Table Headers=
Mode names<br>(Heathwaite) $
Mode names<br>(CA) $
Mixed diatonic<br>modes $
| Table Entries=
Dril $
Dalmatian $
Dorian + Lydian $
Gil $
Galatian $
Aeolian + Lydian $
Kleeth $
Cilician $
Aeolian + Ionian $
Bish $
Bithynian $
Phrygian + Ionian $
Fish $
Pisidian $
Phrygian + Mixolydian $
Jwl $
Illyrian $
Locrian + Mixolydian $
Led $
Lycian $
Locrian + Dorian $
}}


== Theory ==
== Theory ==
Mosh can be thought of as a midpoint between two diatonic scales which are two cyclic orders away from each other. For example, sLsLsLs is the midpoint between the Ionian (major, LLsLLLs) and Phrygian (sLLLsLL) modes. You can prove this by simple addition:
<pre>
  2 2 1 2 2 2 1 (LLsLLLs)
+ 1 2 2 2 1 2 2 (sLLLsLL)
= 3 4 3 4 3 4 3 (sLsLsLs)
</pre>
The rest of the equivalencies are listed in [[#Proposed names]].


=== Low harmonic entropy scales ===
=== Low harmonic entropy scales ===
There are two notable harmonic entropy minima:
There are two notable harmonic entropy minima:


* [[Neutral third scales]], such as dicot, hemififth, and mohajira, in which the generator is a neutral 3rd (around 350¢) and two of them make a 3/2 (702¢).
* [[Neutral third scales]], such as dicot, hemififth, and mohajira, in which the generator is a neutral third (around 350{{c}}) and two of them make a 3/2 (702{{c}}).
* [[Magic]], in which the generator is 5/4 (386¢) and 5 of them make a 3/1 (1902¢).
* [[Magic]], in which the generator is 5/4 (386{{c}}) and five of them make a 3/1 (1902{{c}}), though the step ratios in this range are very hard to the point of being lopsided.


== Tuning ranges ==
== Tuning ranges ==
3\10 represents a dividing line between "neutral third scales" on the bottom (eg. 17edo neutral scale), and scales generated by submajor and major thirds at the top, with 10edo standing in between. The neutral third scales, after three more generators, make MOS [[7L 3s]] (dicoid); the other scales make MOS [[3L 7s]] (sephiroid).
3\10 represents a dividing line between "neutral third scales" on the bottom (eg. 17edo neutral scale), and scales generated by submajor and major thirds at the top, with 10edo standing in between. The neutral third scales, after three more generators, make mos [[7L&nbsp;3s]] (dicoid); the other scales make mos [[3L&nbsp;7s]] (sephiroid).


In dicoid, the generators are all "neutral thirds," and two of them make an approximation of the "perfect fifth." Additionally, the L of the scale is somewhere around a "whole tone" and the s of the scale is somewhere around a "neutral tone".
In dicoid, the generators are all "neutral thirds," and two of them make an approximation of the "perfect fifth." Additionally, the L of the scale is somewhere around a "whole tone" and the s of the scale is somewhere around a "neutral tone".
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=== Ultrasoft ===
=== Ultrasoft ===
[[Ultrasoft]] mosh tunings have step ratios that are less than 4:3, which implies a generator flatter than 7\24 = 350¢.
[[Ultrasoft]] mosh tunings have step ratios that are less than 4:3, which implies a generator flatter than {{nowrap| 7\24 {{=}} 350{{c}} }}.


Ultrasoft mosh can be considered "meantone mosh". This is because the large step is a "meantone" in these tunings, somewhere between near-10/9 (as in [[38edo]]) and near-9/8 (as in [[24edo]]).
Ultrasoft mosh can be considered "meantone mosh". This is because the large step is a "meantone" in these tunings, somewhere between near-10/9 (as in [[38edo]]) and near-9/8 (as in [[24edo]]).


Ultrasoft mosh EDOs include [[24edo]], [[31edo]], [[38edo]], and [[55edo]].
Ultrasoft mosh edos include [[24edo]], [[31edo]], [[38edo]], and [[55edo]].
* [[24edo]] can be used to make large and small steps more distinct (the step ratio is 4/3), or for its nearly pure 3/2.
* [[24edo]] can be used to make large and small steps more distinct (the step ratio is 4/3), or for its nearly pure 3/2.
* [[38edo]] can be used to tune the diminished and perfect mosthirds near [[6/5]] and [[11/9]], respectively.
* [[38edo]] can be used to tune the diminished and perfect mosthirds near [[6/5]] and [[11/9]], respectively.
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| [[11/9]]
| [[11/9]]
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 4\24, 200.00
| 4\24, 200.00
| 5\31, 193.55
| 5\31, 193.55
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| [[9/8]], [[10/9]]
| [[9/8]], [[10/9]]
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 3\24, 150.00
| 3\24, 150.00
| 4\31, 154.84
| 4\31, 154.84
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=== Quasisoft ===
=== Quasisoft ===
Quasisoft tunings of mosh have a step ratio between 3/2 and 5/3, implying a generator sharper than 5\17 = 352.94¢ and flatter than 8\27 = 355.56¢.
Quasisoft tunings of mosh have a step ratio between 3/2 and 5/3, implying a generator sharper than {{nowrap| 5\17 {{=}} 352.94{{c}} }} and flatter than {{nowrap| 8\27 {{=}} 355.56{{c}} }}.


The large step is a sharper major second in these tunings than in ultrasoft tunings. These tunings could be considered "parapyth mosh" or "archy mosh", in analogy to ultrasoft mosh being meantone mosh.
The large step is a sharper major second in these tunings than in ultrasoft tunings. These tunings could be considered "parapyth mosh" or "archy mosh", in analogy to ultrasoft mosh being meantone mosh.
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| 16/13, 11/9
| 16/13, 11/9
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 3\17, 211.76
| 3\17, 211.76
| 5\27, 222.22
| 5\27, 222.22
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| 9/8, 8/7
| 9/8, 8/7
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 2\17, 141.18
| 2\17, 141.18
| 3\27, 133.33
| 3\27, 133.33
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=== Hypohard ===
=== Hypohard ===
Hypohard tunings of mosh have a step ratio between 2 and 3, implying a generator sharper than 3\10 = 360¢ and flatter than 4\13 = 369.23¢.
Hypohard tunings of mosh have a step ratio between 2 and 3, implying a generator sharper than {{nowrap| 3\10 {{=}} 360{{c}} }} and flatter than {{nowrap| 4\13 {{=}} 369.23{{c}} }}.


The large step ranges from a semifourth to a subminor third in these tunings. The small step is now clearly a semitone, ranging from 1\10 (120¢) to 1\13 (92.31¢).
The large step ranges from a semifourth to a subminor third in these tunings. The small step is now clearly a semitone, ranging from 1\10 (120{{c}}) to 1\13 (92.31{{c}}).


The symmetric mode sLsLsLs becomes a distorted double harmonic major in these tunings.
The symmetric mode sLsLsLs becomes a distorted double harmonic major in these tunings.
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| 7\23, 365.22
| 7\23, 365.22
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 2\10, 240.00
| 2\10, 240.00
| 3\13, 276.92
| 3\13, 276.92
| 5\23, 260.87
| 5\23, 260.87
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 1\10, 120.00
| 1\10, 120.00
| 1\13, 92.31
| 1\13, 92.31
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=== Ultrahard ===
=== Ultrahard ===
Ultra tunings of mosh have a step ratio greater than 4/1, implying a generator sharper than 5\16 = 375¢. The generator is thus near a [[5/4]] major third, five of which add up to an approximate [[3/1]]. The 7-note MOS only has two perfect fifths, so extending the chain to bigger MOSes, such as the [[3L 7s]] 10-note MOS, is suggested for getting 5-limit harmony.
Ultra tunings of mosh have a step ratio greater than 4/1, implying a generator sharper than {{nowrap| 5\16 {{=}} 375{{c}} }}. The generator is thus near a [[5/4]] major third, five of which add up to an approximate [[3/1]]. The 7-note mos only has two perfect fifths, so extending the chain to bigger mosses, such as the [[3L&nbsp;7s]] 10-note mos, is suggested for getting 5-limit harmony.


This range is associated with [[magic]] temperament.
This range is associated with [[magic]] temperament.
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| 5/4
| 5/4
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 4\16, 300.00
| 4\16, 300.00
| 5\19, 315.79
| 5\19, 315.79
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| 6/5
| 6/5
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 1\16, 75.00
| 1\16, 75.00
| 1\19, 63.16
| 1\19, 63.16
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|}
|}


== Modes ==
== Scales ==
The various modes of 3L 4s (with [[Modal UDP Notation]] and nicknames coined by [[Andrew Heathwaite]]) are:
* [[Mohaha7]] – 38\131 tuning
 
* [[Neutral7]] – 111\380 tuning
{{MOS modes|Mode Names=dril; gil; kleeth; bish; fish; jwl; led}}
* [[Namo7]] – 128\437 tuning
* [[Rastgross1]] – POTE tuning of [[namo]]
* [[Hemif7]] – 17\58 tuning
* [[Suhajira7]] – POTE tuning of [[suhajira]]
* [[Sephiroth7]] – 9\29 tuning
* [[Magic7]] – 46\145 tuning


== Scale tree ==
== Scale tree ==
Generator ranges:  
Generator ranges:  
* Chroma-positive generator: 342.8571 cents (2\7) to 400 cents (1\3)
* Chroma-positive generator: 342.8571{{c}} (2\7) to 400.0000{{c}} (1\3)
* Chroma-negative generator: 800 cents (2\3) to 857.1429 cents (5\7)
* Chroma-negative generator: 800.0000{{c}} (2\3) to 857.1429{{c}} (5\7)
 
{{MOS tuning spectrum
{| class="wikitable center-all"
| 6/5 = [[Mohaha]] / ptolemy&nbsp;↑
! colspan="6" | Generator
| 5/4 = Mohaha / migration / [[mohajira]]  
! Cents
| 11/8 = Mohaha / mohamaq  
! L
| 7/5 = Mohaha / [[neutrominant]]  
! s
| 10/7 = [[Hemif]] / [[hemififths]]  
! L/s
| 11/7 = [[Suhajira]]  
! Comments
| 13/8 = Golden suhajira (354.8232{{c}})  
|-
| 5/3 = Suhajira / [[ringo]]  
| 2\7 || || || || || || 342.857 || 1 || 1 || 1.000 ||
| 12/7 = [[Beatles]]  
|-
| 13/5 = Unnamed golden tuning (366.2564{{c}})  
| || || || || || 11\38 || 347.368 || 6 || 5 || 1.200 || [[Mohaha]] / ptolemy↑
| 7/2 = [[Sephiroth]]  
|-
| 9/2 = [[Muggles]]  
| || || || || 9\31 || || 348.387 || 5 || 4 || 1.250 || Mohaha / migration / [[mohajira]]
| 5/1 = [[Magic]]  
|-
| 6/1 = [[Würschmidt]]&nbsp;↓
| || || || || || 16\55 || 349.091 || 9 || 7 || 1.286 ||
}}
|-
| || || || 7\24 || || || 350.000 || 4 || 3 || 1.333 ||
|-
| || || || || || 19\65 || 350.769 || 11 || 8 || 1.375 || Mohaha / mohamaq
|-
| || || || || 12\41 || || 351.220 || 7 || 5 || 1.400 || Mohaha / [[neutrominant]]
|-
| || || || || || 17\58 || 351.724 || 10 || 7 || 1.429 || [[Hemif]] / [[Hemififths]]
|-
| || || 5\17 || || || || 352.941 || 3 || 2 || 1.500 ||
|-
| || || || || || 18\61 || 354.098 || 11 || 7 || 1.571 || [[Suhajira]]
|-
| || || || || 13\44 || || 354.545 || 8 || 5 || 1.600 ||
|-
| || || || || || 21\71 || 354.930 || 13 || 8 || 1.625 || Golden suhajira (354.8232¢)
|-
| || || || 8\27 || || || 355.556 || 5 || 3 || 1.667 || Suhajira / [[ringo]]
|-
| || || || || || 19\64 || 356.250 || 12 || 7 || 1.714 || [[Beatles]]
|-
| || || || || 11\37 || || 356.757 || 7 || 4 || 1.750 ||
|-
| || || || || || 14\47 || 357.447 || 9 || 5 || 1.800 ||
|-
| || 3\10 || || || || || 360.000 || 2 || 1 || 2.000 || Basic mosh <br>(Generators smaller than this are proper)
|-
| || || || || || 13\43 || 362.791 || 9 || 4 || 2.250 ||
|-
| || || || || 10\33 || || 363.636 || 7 || 3 || 2.333 ||
|-
| || || || || || 17\56 || 364.286 || 12 || 5 || 2.400 ||
|-
| || || || 7\23 || || || 365.217 || 5 || 2 || 2.500 ||
|-
| || || || || || 18\59 || 366.102 || 13 || 5 || 2.600 || Unnamed golden tuning (366.2564¢)
|-
| || || || || 11\36 || || 366.667 || 8 || 3 || 2.667 ||
|-
| || || || || || 15\49 || 367.347 || 11 || 4 || 2.750 ||
|-
| || || 4\13 || || || || 369.231 || 3 || 1 || 3.000 ||
|-
| || || || || || 13\42 || 371.429 || 10 || 3 || 3.333 ||
|-
| || || || || 9\29 || || 372.414 || 7 || 2 || 3.500 || [[Sephiroth]]
|-
| || || || || || 14\45 || 373.333 || 11 || 3 || 3.667 ||
|-
| || || || 5\16 || || || 375.000 || 4 || 1 || 4.000 ||
|-
| || || || || || 11\35 || 377.143 || 9 || 2 || 4.500 || [[Muggles]]
|-
| || || || || 6\19 || || 378.947 || 5 || 1 || 5.000 || [[Magic]]
|-
| || || || || || 7\22 || 381.818 || 6 || 1 || 6.000 || [[Wuerschmidt]]↓
|-
| 1\3 || || || || || || 400.000 || 1 || 0 || → inf ||
|}


[[Category:Mosh]]
[[Category:Mosh]]
[[Category:7-tone scales]]
[[Category:7-tone scales]]
Retrieved from "https://en.xen.wiki/w/3L_4s"